3.42 \(\int \text {erf}(d (a+b \log (c x^n))) \, dx\)

Optimal. Leaf size=93 \[ x \text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-x \left (c x^n\right )^{-1/n} e^{\frac {1-4 a b d^2 n}{4 b^2 d^2 n^2}} \text {erf}\left (\frac {2 a b d^2+2 b^2 d^2 \log \left (c x^n\right )-\frac {1}{n}}{2 b d}\right ) \]

[Out]

x*erf(d*(a+b*ln(c*x^n)))-exp(1/4*(-4*a*b*d^2*n+1)/b^2/d^2/n^2)*x*erf(1/2*(2*a*b*d^2-1/n+2*b^2*d^2*ln(c*x^n))/b
/d)/((c*x^n)^(1/n))

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Rubi [A]  time = 0.13, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {6397, 2277, 2274, 15, 2276, 2234, 2205} \[ x \text {Erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-x \left (c x^n\right )^{-1/n} e^{\frac {1-4 a b d^2 n}{4 b^2 d^2 n^2}} \text {Erf}\left (\frac {2 a b d^2+2 b^2 d^2 \log \left (c x^n\right )-\frac {1}{n}}{2 b d}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Erf[d*(a + b*Log[c*x^n])],x]

[Out]

x*Erf[d*(a + b*Log[c*x^n])] - (E^((1 - 4*a*b*d^2*n)/(4*b^2*d^2*n^2))*x*Erf[(2*a*b*d^2 - n^(-1) + 2*b^2*d^2*Log
[c*x^n])/(2*b*d)])/(c*x^n)^n^(-1)

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[(a^IntPart[m]*(a*x^n)^FracPart[m])/x^(n*FracPart[m]), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rule 2205

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[(F^a*Sqrt[Pi]*Erf[(c + d*x)*Rt[-(b*Log[F]),
2]])/(2*d*Rt[-(b*Log[F]), 2]), x] /; FreeQ[{F, a, b, c, d}, x] && NegQ[b]

Rule 2234

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[F^(a - b^2/(4*c)), Int[F^((b + 2*c*x)^2/(4*c))
, x], x] /; FreeQ[{F, a, b, c}, x]

Rule 2274

Int[(u_.)*(F_)^((a_.)*(Log[z_]*(b_.) + (v_.))), x_Symbol] :> Int[u*F^(a*v)*z^(a*b*Log[F]), x] /; FreeQ[{F, a,
b}, x]

Rule 2276

Int[(F_)^(((a_.) + Log[(c_.)*(x_)^(n_.)]^2*(b_.))*(d_.))*((e_.)*(x_))^(m_.), x_Symbol] :> Dist[(e*x)^(m + 1)/(
e*n*(c*x^n)^((m + 1)/n)), Subst[Int[E^(a*d*Log[F] + ((m + 1)*x)/n + b*d*Log[F]*x^2), x], x, Log[c*x^n]], x] /;
FreeQ[{F, a, b, c, d, e, m, n}, x]

Rule 2277

Int[(F_)^(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^2*(d_.)), x_Symbol] :> Int[F^(a^2*d + 2*a*b*d*Log[c*x^n] + b^2
*d*Log[c*x^n]^2), x] /; FreeQ[{F, a, b, c, d, n}, x]

Rule 6397

Int[Erf[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)], x_Symbol] :> Simp[x*Erf[d*(a + b*Log[c*x^n])], x] - Dist
[(2*b*d*n)/Sqrt[Pi], Int[1/E^(d*(a + b*Log[c*x^n]))^2, x], x] /; FreeQ[{a, b, c, d, n}, x]

Rubi steps

\begin {align*} \int \text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx &=x \text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {(2 b d n) \int e^{-d^2 \left (a+b \log \left (c x^n\right )\right )^2} \, dx}{\sqrt {\pi }}\\ &=x \text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {(2 b d n) \int \exp \left (-a^2 d^2-2 a b d^2 \log \left (c x^n\right )-b^2 d^2 \log ^2\left (c x^n\right )\right ) \, dx}{\sqrt {\pi }}\\ &=x \text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {(2 b d n) \int e^{-a^2 d^2-b^2 d^2 \log ^2\left (c x^n\right )} \left (c x^n\right )^{-2 a b d^2} \, dx}{\sqrt {\pi }}\\ &=x \text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {\left (2 b d n x^{2 a b d^2 n} \left (c x^n\right )^{-2 a b d^2}\right ) \int e^{-a^2 d^2-b^2 d^2 \log ^2\left (c x^n\right )} x^{-2 a b d^2 n} \, dx}{\sqrt {\pi }}\\ &=x \text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {\left (2 b d x \left (c x^n\right )^{-2 a b d^2-\frac {1-2 a b d^2 n}{n}}\right ) \operatorname {Subst}\left (\int \exp \left (-a^2 d^2+\frac {\left (1-2 a b d^2 n\right ) x}{n}-b^2 d^2 x^2\right ) \, dx,x,\log \left (c x^n\right )\right )}{\sqrt {\pi }}\\ &=x \text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {\left (2 b d e^{\frac {1-4 a b d^2 n}{4 b^2 d^2 n^2}} x \left (c x^n\right )^{-2 a b d^2-\frac {1-2 a b d^2 n}{n}}\right ) \operatorname {Subst}\left (\int \exp \left (-\frac {\left (\frac {1-2 a b d^2 n}{n}-2 b^2 d^2 x\right )^2}{4 b^2 d^2}\right ) \, dx,x,\log \left (c x^n\right )\right )}{\sqrt {\pi }}\\ &=x \text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-e^{\frac {1-4 a b d^2 n}{4 b^2 d^2 n^2}} x \left (c x^n\right )^{-1/n} \text {erf}\left (\frac {2 a b d^2-\frac {1}{n}+2 b^2 d^2 \log \left (c x^n\right )}{2 b d}\right )\\ \end {align*}

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Mathematica [A]  time = 0.26, size = 80, normalized size = 0.86 \[ x \text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-x \text {erf}\left (a d+b d \log \left (c x^n\right )-\frac {1}{2 b d n}\right ) \exp \left (-\frac {\frac {4 a b n-\frac {1}{d^2}}{b^2}+4 n \log \left (c x^n\right )}{4 n^2}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Erf[d*(a + b*Log[c*x^n])],x]

[Out]

x*Erf[d*(a + b*Log[c*x^n])] - (x*Erf[a*d - 1/(2*b*d*n) + b*d*Log[c*x^n]])/E^(((-d^(-2) + 4*a*b*n)/b^2 + 4*n*Lo
g[c*x^n])/(4*n^2))

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fricas [A]  time = 0.48, size = 122, normalized size = 1.31 \[ -\sqrt {b^{2} d^{2} n^{2}} \operatorname {erf}\left (\frac {{\left (2 \, b^{2} d^{2} n^{2} \log \relax (x) + 2 \, b^{2} d^{2} n \log \relax (c) + 2 \, a b d^{2} n - 1\right )} \sqrt {b^{2} d^{2} n^{2}}}{2 \, b^{2} d^{2} n^{2}}\right ) e^{\left (-\frac {4 \, b^{2} d^{2} n \log \relax (c) + 4 \, a b d^{2} n - 1}{4 \, b^{2} d^{2} n^{2}}\right )} + x \operatorname {erf}\left (b d \log \left (c x^{n}\right ) + a d\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erf(d*(a+b*log(c*x^n))),x, algorithm="fricas")

[Out]

-sqrt(b^2*d^2*n^2)*erf(1/2*(2*b^2*d^2*n^2*log(x) + 2*b^2*d^2*n*log(c) + 2*a*b*d^2*n - 1)*sqrt(b^2*d^2*n^2)/(b^
2*d^2*n^2))*e^(-1/4*(4*b^2*d^2*n*log(c) + 4*a*b*d^2*n - 1)/(b^2*d^2*n^2)) + x*erf(b*d*log(c*x^n) + a*d)

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giac [A]  time = 0.62, size = 79, normalized size = 0.85 \[ x \operatorname {erf}\left (b d n \log \relax (x) + b d \log \relax (c) + a d\right ) + \frac {\operatorname {erf}\left (-b d n \log \relax (x) - b d \log \relax (c) - a d + \frac {1}{2 \, b d n}\right ) e^{\left (-\frac {a}{b n} + \frac {1}{4 \, b^{2} d^{2} n^{2}}\right )}}{c^{\left (\frac {1}{n}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erf(d*(a+b*log(c*x^n))),x, algorithm="giac")

[Out]

x*erf(b*d*n*log(x) + b*d*log(c) + a*d) + erf(-b*d*n*log(x) - b*d*log(c) - a*d + 1/2/(b*d*n))*e^(-a/(b*n) + 1/4
/(b^2*d^2*n^2))/c^(1/n)

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maple [F]  time = 0.02, size = 0, normalized size = 0.00 \[ \int \erf \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(erf(d*(a+b*ln(c*x^n))),x)

[Out]

int(erf(d*(a+b*ln(c*x^n))),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {2 \, b d n e^{\left (-b^{2} d^{2} \log \relax (c)^{2} - a^{2} d^{2}\right )} \int \frac {e^{\left (-2 \, b^{2} d^{2} \log \relax (c) \log \left (x^{n}\right ) - b^{2} d^{2} \log \left (x^{n}\right )^{2}\right )}}{{\left (x^{n}\right )}^{2 \, a b d^{2}}}\,{d x}}{\sqrt {\pi } c^{2 \, a b d^{2}}} + x \operatorname {erf}\left (b d \log \left (x^{n}\right ) + {\left (b \log \relax (c) + a\right )} d\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erf(d*(a+b*log(c*x^n))),x, algorithm="maxima")

[Out]

-2*b*d*n*integrate(e^(-b^2*d^2*log(c)^2 - 2*b^2*d^2*log(c)*log(x^n) - b^2*d^2*log(x^n)^2 - 2*a*b*d^2*log(x^n)
- a^2*d^2), x)/(sqrt(pi)*c^(2*a*b*d^2)) + x*erf(b*d*log(x^n) + (b*log(c) + a)*d)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \mathrm {erf}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(erf(d*(a + b*log(c*x^n))),x)

[Out]

int(erf(d*(a + b*log(c*x^n))), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {erf}{\left (d \left (a + b \log {\left (c x^{n} \right )}\right ) \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(erf(d*(a+b*ln(c*x**n))),x)

[Out]

Integral(erf(d*(a + b*log(c*x**n))), x)

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